Ever tried to picture an octagon that isn’t just a flat‑out regular shape, but one that’s been sliced off at the “apex” like a roof?
Worth adding: if you’ve ever stared at a sketch of a stopped‑clock‑hand figure and wondered, “What on earth is the perimeter here? ” you’re not alone.
Most textbooks will hand you a formula and call it a day, but in practice the answer hinges on a few visual clues most people skip. Below is the full‑stack guide to figuring out the perimeter of that oddly‑shaped octagon—step by step, with the shortcuts most teachers never mention Most people skip this — try not to..
Worth pausing on this one It's one of those things that adds up..
What Is the Octagon Below Apex
Picture a regular octagon, the kind you see on stop signs, but then imagine cutting off the top point so the shape now has a flat side where the tip used to be. In geometry lingo that’s an octagon with a truncated apex—basically a nine‑sided figure that still counts as an octagon because two of the original sides merge into the new top edge.
In plain language: you start with eight equal sides, slice off the top corner, and replace it with a new horizontal segment. Plus, the result is still an eight‑sided polygon, just not perfectly regular. The “apex” is that missing corner; the “below apex” part refers to the remaining seven edges that wrap around the rest of the figure.
Why does this matter? Still, because the perimeter isn’t just “8 × side length” anymore. You have to account for the new top edge and the two shortened side edges that meet it And it works..
Why It Matters / Why People Care
Knowing the perimeter of a truncated octagon pops up more often than you think Small thing, real impact..
- Architecture – Roof trusses and decorative cornices sometimes use that shape. A contractor needs the exact fence length to order materials.
- Crafting – When cutting fabric for a quilt block, a mis‑calculated edge can ruin the whole pattern.
- Math contests – A single‑line answer won’t cut it; judges expect you to explain how you handled the missing apex.
If you ignore the cut‑off piece, you’ll end up short by a few inches (or centimeters). On a large scale—say, a stadium’s decorative railing—that adds up to a costly mistake.
How It Works (or How to Do It)
Below is the step‑by‑step method that works for any octagon where the apex has been removed. The key is to break the shape into pieces you already understand: regular sides, a new top edge, and two “trimmed” sides That's the part that actually makes a difference. Surprisingly effective..
1. Identify the original regular octagon dimensions
A regular octagon can be built from a square of side s by cutting off the four corners at a 45° angle. The length of each original side, a, relates to the square’s side s by
[ a = s,( \sqrt{2} - 1 ) ]
If the problem gives you the distance across the shape (the “diameter”) or the length of a side, you can work backwards to find a. Most textbook diagrams label the original side length as L.
2. Measure the height of the cut (the apex removal)
The apex is usually removed by drawing a line parallel to the base at a distance h down from the original corner. That distance is the “depth” of the cut. In many diagrams the new top edge is labeled t (the length of the flat segment that replaces the apex).
If h isn’t given directly, you can often compute it from the angle of the cut. Because the original corners are 45°, the cut creates two right triangles with legs h and h, so the new top edge length is
[ t = 2h ]
(That’s the case when the cut is perfectly horizontal; if it’s slanted, you’ll need a little trigonometry.)
3. Determine the lengths of the two shortened sides
Each side that used to meet at the apex loses a little piece. The original side length a is now split into:
- a remaining segment r that runs from the bottom corner up to the new top edge, and
- a tiny “missing” tip that’s gone.
Because the cut is at 45°, the missing tip length equals h (the same as the vertical drop). So the remaining segment is
[ r = a - h ]
You’ll have two of these r segments—one on each side of the new top edge.
4. Add up all eight sides
Now you have everything you need:
- 5 unchanged sides (the bottom and the four side‑walls that weren’t touched) – each still length a.
- 2 shortened sides – each length r.
- 1 new top edge – length t.
The perimeter P therefore is
[ P = 5a + 2r + t ]
Substituting r = a - h and t = 2h gives a tidy formula that works for any “apex‑cut” octagon:
[ \boxed{P = 7a - 2h} ]
If you know the original side length a and the cut depth h, plug them in and you’ve got the answer That's the part that actually makes a difference. Turns out it matters..
5. Quick sanity check
If h is zero (no cut), the formula collapses to P = 8a, which is exactly the perimeter of a regular octagon.
If h equals a (you cut all the way to the base), the shape becomes a rectangle and the formula reduces to P = 2a + 2a = 4a, which matches the rectangle’s perimeter. Those edge cases confirm the math is sound Most people skip this — try not to..
Common Mistakes / What Most People Get Wrong
- Counting nine sides – The new top edge replaces the missing corner, it doesn’t add an extra side.
- Using the original side length for the trimmed edges – Those two sides shrink by the cut depth; forgetting that throws the perimeter off by 2h.
- Assuming the top edge equals the original side length – Only when the cut depth is exactly half the original side does t equal a. Most diagrams have a shorter top edge.
- Mixing up units – If the diagram gives the diagonal of the original octagon instead of the side, you need to convert first (the diagonal = a · (1 + √2)). Skipping that step leads to a wildly inflated answer.
Practical Tips / What Actually Works
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Sketch it first. Even a rough pencil drawing helps you see which sides stay the same and which get shortened Not complicated — just consistent. Which is the point..
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Label everything. Write a, h, t, r directly on the picture. It forces you to keep track of each piece That's the part that actually makes a difference. But it adds up..
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Double‑check the cut angle. Most textbook octagons use 45°, but a custom diagram might use 30° or 60°. If the angle changes, the relationship t = 2h no longer holds; use basic trigonometry:
[ t = 2h \tan(\theta/2) ]
where θ is the interior angle at the original corner (135° for a regular octagon).
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*Use a calculator for radicals. **Verify with perimeter‑by‑hand.Also, ** The side‑length formula a = s(\sqrt2 - 1) involves a square root; a quick calculator entry avoids rounding errors. ** Add the lengths you measured on the sketch; if the sum matches 7a – 2h, you’re good.
FAQ
Q1: What if the apex is cut off diagonally instead of horizontally?
A: Then the new top edge isn’t parallel to the base. Measure the actual length of that edge (call it t), and measure the two shortened sides directly. The perimeter formula stays P = 5a + 2r + t, but you’ll compute r from the geometry of the cut triangle rather than using a – h The details matter here..
Q2: Can I use this method for a regular octagon with a missing corner that isn’t centered?
A: Yes, as long as the cut removes a single corner. You just need the specific lengths of the two affected sides and the new edge; the other five sides remain unchanged Most people skip this — try not to..
Q3: How do I find a if the diagram only gives the distance across the octagon (the “span”)?
A: The span (corner‑to‑corner distance) of a regular octagon equals a · (1 + √2). Rearrange to get a = span / (1 + √2), then plug that a into the perimeter formula Still holds up..
Q4: Is there a shortcut if the cut depth equals half the original side length?
A: When h = a/2, the perimeter simplifies to P = 7a – a = 6a. So you can just multiply the original side length by six Simple as that..
Q5: Does the formula work for a “truncated” octagon where two opposite corners are removed?
A: Not directly. Removing two corners changes the count of unchanged sides. You’d have to treat each cut separately and adjust the coefficient in front of a accordingly The details matter here..
That’s it. Once you see the shape broken into its original pieces, the perimeter of an octagon with the apex removed is just a matter of plugging a couple of numbers into P = 7a – 2h Still holds up..
Next time you’re staring at a weird‑looking eight‑sided figure, remember: draw, label, and simplify. The answer will come together faster than you think. Happy measuring!
